Integrand size = 24, antiderivative size = 55 \[ \int \frac {5-x}{(3+2 x)^2 \sqrt {2+3 x^2}} \, dx=-\frac {13 \sqrt {2+3 x^2}}{35 (3+2 x)}-\frac {41 \text {arctanh}\left (\frac {4-9 x}{\sqrt {35} \sqrt {2+3 x^2}}\right )}{35 \sqrt {35}} \] Output:
-13*(3*x^2+2)^(1/2)/(105+70*x)-41/1225*35^(1/2)*arctanh(1/35*(4-9*x)*35^(1 /2)/(3*x^2+2)^(1/2))
Time = 0.44 (sec) , antiderivative size = 68, normalized size of antiderivative = 1.24 \[ \int \frac {5-x}{(3+2 x)^2 \sqrt {2+3 x^2}} \, dx=-\frac {13 \sqrt {2+3 x^2}}{35 (3+2 x)}+\frac {82 \text {arctanh}\left (\frac {3 \sqrt {3}+2 \sqrt {3} x-2 \sqrt {2+3 x^2}}{\sqrt {35}}\right )}{35 \sqrt {35}} \] Input:
Integrate[(5 - x)/((3 + 2*x)^2*Sqrt[2 + 3*x^2]),x]
Output:
(-13*Sqrt[2 + 3*x^2])/(35*(3 + 2*x)) + (82*ArcTanh[(3*Sqrt[3] + 2*Sqrt[3]* x - 2*Sqrt[2 + 3*x^2])/Sqrt[35]])/(35*Sqrt[35])
Time = 0.18 (sec) , antiderivative size = 55, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.125, Rules used = {679, 488, 219}
Below are the steps used by Rubi to obtain the solution. The rule number used for the transformation is given above next to the arrow. The rules definitions used are listed below.
\(\displaystyle \int \frac {5-x}{(2 x+3)^2 \sqrt {3 x^2+2}} \, dx\) |
\(\Big \downarrow \) 679 |
\(\displaystyle \frac {41}{35} \int \frac {1}{(2 x+3) \sqrt {3 x^2+2}}dx-\frac {13 \sqrt {3 x^2+2}}{35 (2 x+3)}\) |
\(\Big \downarrow \) 488 |
\(\displaystyle -\frac {41}{35} \int \frac {1}{35-\frac {(4-9 x)^2}{3 x^2+2}}d\frac {4-9 x}{\sqrt {3 x^2+2}}-\frac {13 \sqrt {3 x^2+2}}{35 (2 x+3)}\) |
\(\Big \downarrow \) 219 |
\(\displaystyle -\frac {41 \text {arctanh}\left (\frac {4-9 x}{\sqrt {35} \sqrt {3 x^2+2}}\right )}{35 \sqrt {35}}-\frac {13 \sqrt {3 x^2+2}}{35 (2 x+3)}\) |
Input:
Int[(5 - x)/((3 + 2*x)^2*Sqrt[2 + 3*x^2]),x]
Output:
(-13*Sqrt[2 + 3*x^2])/(35*(3 + 2*x)) - (41*ArcTanh[(4 - 9*x)/(Sqrt[35]*Sqr t[2 + 3*x^2])])/(35*Sqrt[35])
Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(1/(Rt[a, 2]*Rt[-b, 2]))* ArcTanh[Rt[-b, 2]*(x/Rt[a, 2])], x] /; FreeQ[{a, b}, x] && NegQ[a/b] && (Gt Q[a, 0] || LtQ[b, 0])
Int[1/(((c_) + (d_.)*(x_))*Sqrt[(a_) + (b_.)*(x_)^2]), x_Symbol] :> -Subst[ Int[1/(b*c^2 + a*d^2 - x^2), x], x, (a*d - b*c*x)/Sqrt[a + b*x^2]] /; FreeQ [{a, b, c, d}, x]
Int[((d_.) + (e_.)*(x_))^(m_)*((f_.) + (g_.)*(x_))*((a_) + (c_.)*(x_)^2)^(p _.), x_Symbol] :> Simp[(-(e*f - d*g))*(d + e*x)^(m + 1)*((a + c*x^2)^(p + 1 )/(2*(p + 1)*(c*d^2 + a*e^2))), x] + Simp[(c*d*f + a*e*g)/(c*d^2 + a*e^2) Int[(d + e*x)^(m + 1)*(a + c*x^2)^p, x], x] /; FreeQ[{a, c, d, e, f, g, m, p}, x] && EqQ[Simplify[m + 2*p + 3], 0]
Time = 0.78 (sec) , antiderivative size = 50, normalized size of antiderivative = 0.91
method | result | size |
risch | \(-\frac {13 \sqrt {3 x^{2}+2}}{35 \left (2 x +3\right )}-\frac {41 \sqrt {35}\, \operatorname {arctanh}\left (\frac {2 \left (4-9 x \right ) \sqrt {35}}{35 \sqrt {12 \left (x +\frac {3}{2}\right )^{2}-36 x -19}}\right )}{1225}\) | \(50\) |
default | \(-\frac {13 \sqrt {3 \left (x +\frac {3}{2}\right )^{2}-9 x -\frac {19}{4}}}{70 \left (x +\frac {3}{2}\right )}-\frac {41 \sqrt {35}\, \operatorname {arctanh}\left (\frac {2 \left (4-9 x \right ) \sqrt {35}}{35 \sqrt {12 \left (x +\frac {3}{2}\right )^{2}-36 x -19}}\right )}{1225}\) | \(53\) |
trager | \(-\frac {13 \sqrt {3 x^{2}+2}}{35 \left (2 x +3\right )}+\frac {41 \operatorname {RootOf}\left (\textit {\_Z}^{2}-35\right ) \ln \left (\frac {9 \operatorname {RootOf}\left (\textit {\_Z}^{2}-35\right ) x -4 \operatorname {RootOf}\left (\textit {\_Z}^{2}-35\right )+35 \sqrt {3 x^{2}+2}}{2 x +3}\right )}{1225}\) | \(66\) |
Input:
int((5-x)/(2*x+3)^2/(3*x^2+2)^(1/2),x,method=_RETURNVERBOSE)
Output:
-13/35*(3*x^2+2)^(1/2)/(2*x+3)-41/1225*35^(1/2)*arctanh(2/35*(4-9*x)*35^(1 /2)/(12*(x+3/2)^2-36*x-19)^(1/2))
Time = 0.11 (sec) , antiderivative size = 74, normalized size of antiderivative = 1.35 \[ \int \frac {5-x}{(3+2 x)^2 \sqrt {2+3 x^2}} \, dx=\frac {41 \, \sqrt {35} {\left (2 \, x + 3\right )} \log \left (-\frac {\sqrt {35} \sqrt {3 \, x^{2} + 2} {\left (9 \, x - 4\right )} + 93 \, x^{2} - 36 \, x + 43}{4 \, x^{2} + 12 \, x + 9}\right ) - 910 \, \sqrt {3 \, x^{2} + 2}}{2450 \, {\left (2 \, x + 3\right )}} \] Input:
integrate((5-x)/(3+2*x)^2/(3*x^2+2)^(1/2),x, algorithm="fricas")
Output:
1/2450*(41*sqrt(35)*(2*x + 3)*log(-(sqrt(35)*sqrt(3*x^2 + 2)*(9*x - 4) + 9 3*x^2 - 36*x + 43)/(4*x^2 + 12*x + 9)) - 910*sqrt(3*x^2 + 2))/(2*x + 3)
\[ \int \frac {5-x}{(3+2 x)^2 \sqrt {2+3 x^2}} \, dx=- \int \frac {x}{4 x^{2} \sqrt {3 x^{2} + 2} + 12 x \sqrt {3 x^{2} + 2} + 9 \sqrt {3 x^{2} + 2}}\, dx - \int \left (- \frac {5}{4 x^{2} \sqrt {3 x^{2} + 2} + 12 x \sqrt {3 x^{2} + 2} + 9 \sqrt {3 x^{2} + 2}}\right )\, dx \] Input:
integrate((5-x)/(3+2*x)**2/(3*x**2+2)**(1/2),x)
Output:
-Integral(x/(4*x**2*sqrt(3*x**2 + 2) + 12*x*sqrt(3*x**2 + 2) + 9*sqrt(3*x* *2 + 2)), x) - Integral(-5/(4*x**2*sqrt(3*x**2 + 2) + 12*x*sqrt(3*x**2 + 2 ) + 9*sqrt(3*x**2 + 2)), x)
Time = 0.14 (sec) , antiderivative size = 53, normalized size of antiderivative = 0.96 \[ \int \frac {5-x}{(3+2 x)^2 \sqrt {2+3 x^2}} \, dx=\frac {41}{1225} \, \sqrt {35} \operatorname {arsinh}\left (\frac {3 \, \sqrt {6} x}{2 \, {\left | 2 \, x + 3 \right |}} - \frac {2 \, \sqrt {6}}{3 \, {\left | 2 \, x + 3 \right |}}\right ) - \frac {13 \, \sqrt {3 \, x^{2} + 2}}{35 \, {\left (2 \, x + 3\right )}} \] Input:
integrate((5-x)/(3+2*x)^2/(3*x^2+2)^(1/2),x, algorithm="maxima")
Output:
41/1225*sqrt(35)*arcsinh(3/2*sqrt(6)*x/abs(2*x + 3) - 2/3*sqrt(6)/abs(2*x + 3)) - 13/35*sqrt(3*x^2 + 2)/(2*x + 3)
Leaf count of result is larger than twice the leaf count of optimal. 125 vs. \(2 (44) = 88\).
Time = 0.14 (sec) , antiderivative size = 125, normalized size of antiderivative = 2.27 \[ \int \frac {5-x}{(3+2 x)^2 \sqrt {2+3 x^2}} \, dx=\frac {1}{2450} \, \sqrt {35} {\left (13 \, \sqrt {35} \sqrt {3} + 82 \, \log \left (\sqrt {35} \sqrt {3} - 9\right )\right )} \mathrm {sgn}\left (\frac {1}{2 \, x + 3}\right ) - \frac {41 \, \sqrt {35} \log \left (\sqrt {35} {\left (\sqrt {-\frac {18}{2 \, x + 3} + \frac {35}{{\left (2 \, x + 3\right )}^{2}} + 3} + \frac {\sqrt {35}}{2 \, x + 3}\right )} - 9\right )}{1225 \, \mathrm {sgn}\left (\frac {1}{2 \, x + 3}\right )} - \frac {13 \, \sqrt {-\frac {18}{2 \, x + 3} + \frac {35}{{\left (2 \, x + 3\right )}^{2}} + 3}}{70 \, \mathrm {sgn}\left (\frac {1}{2 \, x + 3}\right )} \] Input:
integrate((5-x)/(3+2*x)^2/(3*x^2+2)^(1/2),x, algorithm="giac")
Output:
1/2450*sqrt(35)*(13*sqrt(35)*sqrt(3) + 82*log(sqrt(35)*sqrt(3) - 9))*sgn(1 /(2*x + 3)) - 41/1225*sqrt(35)*log(sqrt(35)*(sqrt(-18/(2*x + 3) + 35/(2*x + 3)^2 + 3) + sqrt(35)/(2*x + 3)) - 9)/sgn(1/(2*x + 3)) - 13/70*sqrt(-18/( 2*x + 3) + 35/(2*x + 3)^2 + 3)/sgn(1/(2*x + 3))
Time = 6.02 (sec) , antiderivative size = 53, normalized size of antiderivative = 0.96 \[ \int \frac {5-x}{(3+2 x)^2 \sqrt {2+3 x^2}} \, dx=\frac {41\,\sqrt {35}\,\ln \left (x+\frac {3}{2}\right )}{1225}-\frac {41\,\sqrt {35}\,\ln \left (x-\frac {\sqrt {3}\,\sqrt {35}\,\sqrt {x^2+\frac {2}{3}}}{9}-\frac {4}{9}\right )}{1225}-\frac {13\,\sqrt {3}\,\sqrt {x^2+\frac {2}{3}}}{70\,\left (x+\frac {3}{2}\right )} \] Input:
int(-(x - 5)/((2*x + 3)^2*(3*x^2 + 2)^(1/2)),x)
Output:
(41*35^(1/2)*log(x + 3/2))/1225 - (41*35^(1/2)*log(x - (3^(1/2)*35^(1/2)*( x^2 + 2/3)^(1/2))/9 - 4/9))/1225 - (13*3^(1/2)*(x^2 + 2/3)^(1/2))/(70*(x + 3/2))
Time = 0.28 (sec) , antiderivative size = 83, normalized size of antiderivative = 1.51 \[ \int \frac {5-x}{(3+2 x)^2 \sqrt {2+3 x^2}} \, dx=\frac {-455 \sqrt {3 x^{2}+2}+82 \sqrt {35}\, \mathrm {log}\left (\sqrt {3 x^{2}+2}\, \sqrt {35}+9 x -4\right ) x +123 \sqrt {35}\, \mathrm {log}\left (\sqrt {3 x^{2}+2}\, \sqrt {35}+9 x -4\right )-82 \sqrt {35}\, \mathrm {log}\left (2 x +3\right ) x -123 \sqrt {35}\, \mathrm {log}\left (2 x +3\right )}{2450 x +3675} \] Input:
int((5-x)/(3+2*x)^2/(3*x^2+2)^(1/2),x)
Output:
( - 455*sqrt(3*x**2 + 2) + 82*sqrt(35)*log(sqrt(3*x**2 + 2)*sqrt(35) + 9*x - 4)*x + 123*sqrt(35)*log(sqrt(3*x**2 + 2)*sqrt(35) + 9*x - 4) - 82*sqrt( 35)*log(2*x + 3)*x - 123*sqrt(35)*log(2*x + 3))/(1225*(2*x + 3))